The cone

Take a square piece of graph paper, label the horizontal axis "Time" and the vertical axis "Length". We will be using natural units so one grid line in the horizontal axis is one natural unit of time, and one grid line in the horizontal is one natural unit of space. Now draw a line on the graph, such that it represents the speed of light....one unit of length for one unit of time. The line will be at 45 degrees from either axis. Color everything below the line blue, for any line drawn from the origin to any point in that region will represent a velocity faster than the speed of light. Color anything above the 45 degree line red, for any line from the origin to any point in that region will be slower than light speed.

Now roll the paper into a cone with the origin at the pointy end. You see we have made a structure in which the axis representing time is joined to the axis representing space. We can do this and be meaningful because Albert introduced us to the idea that space and time are the same thing.

We have made a spacetime cone. You can readily see that it is expanding from the point of origin. You can also see that half of it is blue, and half is red. Now look at the line that we called the speed of light. It was on the diagonal of the graph when the paper was flat, so it is longer than the two axies, and it sticks out farther at the open end of the cone.

The open end of the cone describes a shape which I am going to say is a hyperbola, although I am forced to admit that I am reaching into my memory a ways for this information and it may be suspect to correction. Still, some little kid in me is jumping up and down, insisting that it must be so, or that, even if it isn't a hyperbola (that is a grown up word that I use, but the kid isn't really sure what it means), even if it isn't a hyperbola, it must be something really important, and deserving of a big fine grown up word with lots of letters and syllables.

Well, then, if we consider the idea of an object at rest, and draw the line that object must take on the graph, it is of course nothing more than the length axis, which we now see is really the same as the time axis. It is at the lowest, smoothest part of the open end of the cone. The line representing the speed of light is on the opposite, pointy side of the cone.

Now here is a curious thing. Any object at rest at the origin travels straight up the short side of the cone. Any object at rest from a point after the origin has to follow a spirol path. If you look at the grid lines, one of which the object at rest from a position after the orgin has to follow, you can clearly see that they spirol around the cone. But, curved as they are, they never make it all the way around!

Oh well, if this is the cone of the universe, with the big bang at the origin, then we are at a place far, far from the pointy end, and out in the middle somewhere where things seem pretty flat again. We might as well not have rolled the paper up at all, looking at things on our scale in space and time.

But here is something. Everything that has an origin has a spacetime cone like this. If we look very close to the origin, we can see the unity of spacetime. Usually we can't look that close, but maybe there are some cases where we get a little glimpse. Say, in a cloud chamber where a gamma photon decays into a pair of 'trons, one an electron and one a positron. I have just seen a pretty picture of that today, I'll try to find the link,

but I am sure you have seen it too. It has a very pointy end where the gamma photon decays, which goes into two spirols, one above the other, curling in opposite directions.

Something about that picture. It is taken in a strong magnetic field, of course, which is what causes the particles to curl around like that. But what I want to know is, what happens to the particals, once they get to the center of the spirol? Where do they go then? Are they in a rest frame after that? Or do they, as it seems from their tracks, simply dissappear?

Oh well, just thought I'd plunk down my uncertainty, and see if anyone wants to make a play on it.

Take a square piece of graph paper, label the horizontal axis "Time" and the vertical axis "Length". We will be using natural units so one grid line in the horizontal axis is one natural unit of time, and one grid line in the horizontal is one natural unit of space. Now draw a line on the graph, such that it represents the speed of light....one unit of length for one unit of time. The line will be at 45 degrees from either axis. Color everything below the line blue, for any line drawn from the origin to any point in that region will represent a velocity faster than the speed of light. Color anything above the 45 degree line red, for any line from the origin to any point in that region will be slower than light speed.

I think you've got it backwards, Richard. With the time axis horizantal, the lower wedge of speeds are slower than light ("Timelike" is the buzzword). The ones above are faster than light, or "spacelike". It's easy on diagrams like the one you have drawn; however you draw your time and space axes, the lines representing lightspeed ("lightlike speeds" = "the lightcone") are halfway between, the timelike speeds are in the wedge next to your time axis and the spacelike ones are in the wedge next to your space axis. This works even if the time and space axis make an acute angle with each other, as in the results of a Lorents transform. Take 87% of c, between an original rectilinear frame and another one. At that speed the conversion factor is 1/2, so the space axis will be rotated so that 1 unit on the original axis projects to 1/2 unit on the new one. And similarly for the time axis. These new axes will then be at an acute angle to each other, but you can still construct the lines for lightlike, spacelike and timelike speeds according to the construction above.

I hadn't thought of the Lorentz transform thing. It is pretty hard to draw angles on cones. Luckily we are in a very flat region far out from the origin of everything, so we can neglect the curve mostly. But what happens near the origin? Something to think about.

Your cone is a hyperbola, and its curvature is zero. The reason it has zero curvature is that you can make it out of a flat rectangle; trying doing that with a sphere (which has uniform positive curvature).

Actually, I had some more thoughts about the cone, but didn't post them because the image got too complex for any conclusions, and I decided I was pushing the analogy too far. Basically, the question is, what happens when new particles originate? When spacetime gets rolled up into a cone for them, doesn't neighboring spacetime get pinched? Nah, that has to be stretching too far.

As for the sphere, well, I haven't tried it yet, but you know it seems to me that a section cut from say a large rubber ball could be rolled into a sort of cone shape. In fact, if you took a large rubber ball and cut it into two halves along an equator, then cut a slit up to the pole on a longitude, it seems fairly clear to me that a sphereoid analog to the cone could be formed by overlapping the edges. The shape of the equatorial cut would deform into a near-circle with a pointy dent in it, at right angles to the plane of the circle.

Richard, if you cut a sphere (And I mean just the surface, not the solid, which topologists call a Ball), into two simple pieces, each piece is topologically a disk, and can easily be deformed flat. But the whole uncut sphere cannot - basic topological theorem! Not Mercator or Einstein or anybody can build a one-to-one continuous map between a spherical surface and a flat one; it has to blow up somewhere.

The same thing would work for a spherical hypersurface if spacetime were shaped that way. It might look flat in one place, but it couldn't get flat everywhere (I am talking topology here, not metric geometry, so on a SMALL scale it could look flat at each point, just as the Earth does, geography apart. But overall there has to be some exception to overall flatness with a topological sphere.

With a hyperboloid not so. Oddly, although it looks even more curved to us, it can be mapped one-to-one, with all converging sequences going into converging sequences, into a plane.

In topological terms, is a circle the same as a line? If we flatten the circle edgewise, we get a two sided line, so I guess the answer is no. A line has only one side. A circle deformed into a line has two lines, placed side by side, but without glue they must always remain separate.

Zeno's Shade!

A line cannot be equivalent to a circle because it is forbidden to glue the two ends together.

Is a point equivalent to a line? Can we shrink the line down to a single point? I suppose a line must contain the two special case endpoints, so it can never be shrunk down to a single point. And we can't cut off the end points, leaving the next interior point to be the end. The endpoints of a line really must be special cases, identifiable as different from all the other points of the line.

What happens if we shrink the circle down to the fewest possible points? The circle has no special case points like the end points of the line. My geometry leads me to believe a circle must have at least three points, but any three points of the circle will do.

Just as in geometry there is always a point between two points, in topology we can always remove a point between two points. But we are forbidden to cut off an endpoint.

A reduced circle then consists of three intersecting lines and three vertices, which, once we have reduced the circle to a minimum, become special. You could name these vertices, A, B, and C, and following the rule for the endpoints of the line, you could say these points can never be cut off or removed. Every circle must have these three points. And then there are the associated tangent or secant lines, from trigonometry, AB, BC, and CA. Here CA is the same as AC. These lines can never cross to form a new intersection.

If we reduce a sphere to its mimimum number of points, there must remain four points, and again we might argue from the special case of the endpoints of the line that these four points are identifiable and can be named, A,B,C, and D. Every sphere must have these four special points, which can be anywhere on the surface of the sphere, but which can never be removed.

There is one other point which is not part of the sphere but which is part of the definition of the sphere, and that is the origin. Since it is not part of the sphere it does not act in a way to hold the four points of the reduced sphere apart, but it is still definable and always present in any spherical form. Hence we can define four antipodal points on the sphere, as those points which are opposite the four named points. These eight points can then be placed equidistant from each other, to form a cube within the sphere. So we see a sphere is topologically the same as a cube.

Now I have a special fondness for the cubeoctahedron, which is the kernal of the isomatrix and can be formed intuitively from stacked spheres. So naturally I want to transmogrify the eight points defined above into twelve points, which is the number of contact points between spheres in the Kepler stack. Evidently I need to find a way to add another four special, named points to the sphere. So far I have A,A',B,B',C,C',D, and D', where ABC and D were obtained by reducing the sphere to minimum points, and the related primes were generated by expanding the sphere and taking a line through the origin from each of the minimum points. I defined the eight points so named as equidistant on the surface of the sphere to form the cube.

I haven't thought of a way to generate the twelve points from the eight points yet. Maybe the twelve points cannot be generated internally, but require the concept of stacking multiple spheres to attain definition. I should like to prove that it is possible to generate the twelve points from the eight points, or else to prove that it is not possible without outside help from Kepler. If it is not possible it seems to me that there is a curious disjoint here. We would need multiple spheres, and hence multiple origins, to form our percieved space.

That seems to me to translate into multiple origins for our common spacetime. Many little bangs instead of one Big Bang?

I am going to go have a nice, long think. Be back later, if the little bangers don't end up going off in a huff and refuse to talk to each other.

You take the cube constructed in the sphere as in the last post, and notice that a circle can be defined in the sphere by using four points on diagonal corners of the cube. This involves using two named points and their opposites. The same can be done using the other two named points and their opposites. These two circles each contain four named points and two intersections. Then do the same thing again, but this time use one named point on one of the circles just drawn, and one named point from the other circle. This action can be repeated once more, so that now the sphere is marked by four circles, each of which contains four named points, and each of which also contains two intersection points, not previously named. This makes the twelve points, equidistant from each other on the surface of the cube.

The lake is breaking up, waves pushing ice blocks up on shore. I am going down to listen to the sound this makes, a musical sound, like ice cubes sloshing in a glass of water.

"We give an overview of the current issues in early universe cosmology and consider the potential resolution of these issues in an as yet nascent spin foam cosmology. The model is the Barrett-Crane Model for quantum gravity along with a generalization of manifold complexes to complexes including conical singularities."

You may wish to glance at the pictures here! especially figure 3. which has some cones in it. there is an earlier paper by Louis Crane solo which I think also treats his "conical matter proposal"

http://arxiv.org/gr-qc/0110060 [Broken] A new approach to the geometrization of matter
L. Crane

In no way can I personally approve or advocate anything in these papers, understanding of which I am as innocent as a newborn babe. for indeed I have not the slightest clue as to what they are about, except that they are about cones.